Browne — On an Integral Equation proposed by Abel. 65 



has a meaning. Let us take m an even number, equal to 2r. We have 



t*+i<>K{t)clt 



e - « (i * «)Z (e-«) cos tiu du 



1 "18 



tScos (6 log t)K(t)dt 



J 



1 t* sin (0 log*) ■£"(*) *T 



J 



«-«(!♦ s) K{e- U ) sin 0% dw 



If -£"(£) remains finite for < t < 1, we can show that § has a meaning for 



m = 2. Putting 



e -«(l + s)X(e-») = P(u), 

 we have 



Lt 



Lt 



, Lt 



-I 



■ t r r ( 



P (u) cos 6u du 



dO 



P (u) P (v) cos du cos 8v du dv 



d6 



Lt 



J=CC 



P (m) P («;) ] cos (w + v) + cos (% - ») J du dv 



=° p . sin £(« + ») , 



P (m) P (») — — du dv, 



dd 



o J o 



u + v 



Lt 



'fWfW^^M 



.jo u - *> 



since the function P is absolutely integrable from to oo. 

 Putting u =u, u + v = w, we have 



Lt 



Lt 



{=» 



1 sin £ (u + v) 



P (u) P (v) — - dudv. 



U + V 



, sm %w 



dw 



w 



P (u) P (w - u) du, 



. Lt I"" EL5?j»( w )a„ where P(0) = 



= 0. 

 Similarly, putting « = v, u - v = w, we have 



P («) P («) '- du dv 



Lt 



f = cc 



Lt 



<- +0 ° , sin ?;r 

 w 



[P(v)fdv, 



P(v) P{v + w)dv 



